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JOURNAL OFSOUND ANDVIBRATION
Journal of Sound and Vibration 290 (2006) 223241
Modal analysis and shape optimization of rotatingcantilever
beams
Abstract
ARTICLE IN PRESS
www.elsevier.com/locate/jsvi
Corresponding author. Tel.: +82 2 2290 0446; fax: +82 2 2293
0570.0022-460X/$ - see front matter r 2005 Elsevier Ltd. All rights
reserved.
doi:10.1016/j.jsv.2005.03.014
E-mail addresses: [email protected] (H.H. Yoo),
[email protected] (J.E. Cho),
[email protected] (J. Chung).When a cantilever beam rotates
about an axis perpendicular to its neutral axis, its modal
characteristicsoften vary signicantly. If the geometric shape and
the material property of the beam are given, the
modalcharacteristic variations can be accurately estimated
following a well-established analysis procedureemploying assumed
mode method or nite element method. In many practical design
situations, however,some modal characteristics are usually specied
as design requirements and the geometric shape thatsatises the
requirements needs to be found. In the present study, certain modal
characteristic requirementssuch as maximum or minimum slope natural
frequency loci are specied and the geometric shapes thatsatisfy the
requirements are obtained through an optimization procedure.r 2005
Elsevier Ltd. All rights reserved.Hong Hee Yooa,, Jung Eun Chob,
Jintai Chungc
aSchool of Mechanical Engineering, Hanyang University, 17
Haengdang-Dong Sungdong-Gu, Seoul 133-791,
Republic of KoreabR&D Center Chassis Team, Hyundai Motor
Company, Jangduk-Dong, Whasung-Si, Kyunggi-Do 445-706,
Republic of KoreacDepartment of Mechanical Engineering, Hanyang
University, 1271 Sa-1-Dong, Ansan, Kyunggi-Do 425-791,
Republic of Korea
Received 12 February 2004; received in revised form 9 March
2005; accepted 26 March 2005
Available online 21 June 2005
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designs of the rotating structures.Southwell and Gough [1]
pioneered to investigate the modal characteristics of rotating
cantilever beams in 1920s, and their monumental work was
followed by many theoretical and
ARTICLE IN PRESSnumerical studies (see, for instance, Refs.
[25]). More recently, advanced methods weredeveloped and more
complicated effects (see, for instance, Refs. [612]) were
considered toanalyze the modal characteristics of rotating
structures. Using these methods, the modalcharacteristics of
rotating cantilever structures could be effectively analyzed if the
geometric andthe material properties are given. However, in many
practical design situations, the geometricshapes of rotating
structures need to be found (instead of being given) while their
modalcharacteristics are specied as design requirements (to avoid
undesirable vibration problems).Despite the best effort of the
authors, research works on such inverse problems were rarely
foundin literature.The purpose of the present study is to nd the
optimal shapes of rotating cantilever beams that
provide some specic modal characteristics. Maximal or minimal
increasing rate of a naturalfrequency versus the angular speed
could be one of the specic modal characteristics. In thepresent
study, the cross-section of the rotating beam is assumed
rectangular and the length of thebeam is divided into multiple
segments. The thickness and the width at every segment are
assumedto be cubic spline functions. The stage (the segments ends)
values of the thickness and the widthare employed as design
variables and optimization problems that include the design
requirementsof specic modal characteristics are formulated. An
optimization method that combines a geneticalgorithm [13] along
with a gradient-based search algorithm [14] is employed to solve
theproblems in the present study.
2. Derivation of the modal equations of a rotating cantilever
beam
In the present study, a linear dynamic modeling method that
employs hybrid deformationvariables (see Ref. [11]) is utilized to
derive the equations of motion for rotating cantilever beams.The
following assumptions are employed. Firstly, the beam has
homogeneous and isotropicmaterial properties. Secondly, the beam
has slender shape so that shear and rotary inertia effectsare
ignored. Finally, the stretching and the out-of-plane bending
deformations are onlyconsidered. These assumptions are made to
simplify the modeling procedure and to focus on themajor interest
of the present study, that is how to nd the optimal shape of
rotating beams that1. Introduction
Cantilever beam-like structures can be found in many practical
engineering examples. To designsuch structures, it is necessary to
calculate natural frequencies to avoid undesirable problems suchas
resonance phenomena. It is a common practice to nd the natural
frequencies of stationarystructures if their geometric and material
properties are given. Some cantilever structures (forinstance,
turbine and helicopter blades), however, rotate during their normal
operation. Due tothe rotational motion, the modal characteristics
of cantilever structures often vary signicantly.Therefore, the
variations of modal characteristics need to be estimated accurately
for reliable
H.H. Yoo et al. / Journal of Sound and Vibration 290 (2006)
223241224satisfy certain design requirements of modal
characteristics.
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H.H. Yoo et al. / Journal of Sound and Vibration 290 (2006)
223241 225Fig. 1. Conguration of a rotating cantilever beam.Fig. 1
shows the conguration of a cantilever beam xed to a rigid hub
(reference frame A) thatrotates with constant angular speed O about
the axis of a^3. In the gure, a^1; a^2, and a^3 representorthogonal
unit vectors xed in the rigid hub; ~u denotes the elastic
deformation of a generic point;u and w denote the axial and the
out-of-plane bending deformation components, respectively; ands
denotes the arc-length stretch (stretch along the deformed beam
axis). The angular velocity ofthe rigid hub A and the velocity of
point O can be expressed as follows:
~oA Oa^3; ~vO rOa^2, (1)where r denotes the radius of the rigid
hub. Then the velocity of the generic point P can be derivedas
follows:
~vP _ua^1 Or x ua^2 _wa^3, (2)where x is the distance from point
O to the generic point in the un-deformed conguration.In the
present work, s and w are approximated (by employing the assumed
mode method) as
follows:
sx; t Xm1j1
f1jxq1jt,
wx; t Xm2j1
f2jxq2jt, (3)
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ARTICLE IN PRESSwhere f1j and f2j are mode functions, q1j and
q2j are generalized coordinates and m1 and m2 are thenumbers of the
generalized coordinates. Since the non-Cartesian variable s is
approximated, thefollowing geometric relation needs to be employed
to derive the equations of motion.
x s Z x0
1 quqs
2 qw
qs
2" #1=2ds. (4)
Using a binomial expansion of Eq. (4)
s u 12
Z x0
qwqs
2ds higher degree terms. (5)
Differentiation of the above equation with respect to time
gives
_s _u Z x0
qwqs
q _wqs
ds higher degree terms. (6)
By substituting Eqs. (5) and (6) (while neglecting the higher
degree terms) into Eq. (2), the velocityof the generic point P can
be obtained as follows:
~vP _s Z x0
qwqs
q _wqs
ds
a^1 O r x s
1
2
Z x0
qwqs
2ds
!" #a^2 _wa^3. (7)
The partial derivatives of the velocity of P with respect to the
generalized speeds ( _q1i and _q2i) canbe obtained as follows:
q~vP
q _q1i f1ia^1 i 1; 2; . . . ;m1;
q~vP
q _q2i
Xm2j1
Z x0
f2i;sf2j;s ds
q2j
" #a^1 f2ia^3 i 1; 2; . . . ;m2:
(8)
Later, Eq. (8) will be employed to obtain the generalized
inertia forces in the equations of motion.Now, to obtain the
generalized active forces in the equations of motion, one needs the
strain
energy expression of the beam, which is expressed as
follows:
U 12
Z L0
EAqsqx
2 EIyy
q2wqx2
2" #dx, (9)
where L denotes the beam length, E denotes Youngs modulus, A
denotes the cross-sectional areaand Iyy denotes the second area
moment of the cross-section. With the assumption of
neglectingrotary inertia effect, the equations of motion can be
obtained from the following equation:Z L
0
rAq~vP
q _qi
~aP dx qU
qqi 0 i 1; 2; . . . ; m, (10)
where r denotes the density, qi consists of q1i and q2i, m is
the sum of m1 and m2 and ~aP denotes
the acceleration of the generic point which can be obtained by
differentiating the velocity
H.H. Yoo et al. / Journal of Sound and Vibration 290 (2006)
223241226expression given in Eq. (7). Employing Eq. (10), the
equations of motion can be nally derived
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H.H. Yoo et al. / Journal of Sound and Vibration 290 (2006)
223241 227as follows:
Xm1j1
Z L0
rAf1if1j dx
q1j Xm1j1
O2Z L0
rAf1if1j dx
Z L0
EAf1i;xf1j;x dx
q1j
rO2Z L0
rAf1i dx
O2Z L0
rAxf1i dx
0 i 1; 2; . . . ;m1, 11
Xm2j1
Z L0
rAf2if2j dx
q2j Xm2j1
Z L0
EIyyf2i;xxf2j;xx dx
q2j
Xm2j1
O2Z L0
rAxZ x0
f2i;sf2j;s ds
dx
q2j
Xm2j1
rO2Z L0
rAZ x0
f2i;sf2j;s ds
dx
q2j
0 i 1; 2; . . . ;m2. 12As can be observed from Eqs. (11) and
(12), the bending equations are decoupled from thestretching
equations. Since natural frequencies of bending modes are much
lower than those ofstretching modes, only the bending equations
will be employed for the modal analysis. Usingmatrix notation, the
bending equations can be expressed as follows:
Mf q2g hKB O2KGifq2g 0, (13)where the elements of M, KB, and KG
are dened as follows:
Mij Z L0
rbhf2if2j dx, (14)
KBij Z L0
Ebh3
12f2i;xxf2j;xx dx, (15)
KGij Z L0
rbhx rZ x0
f2i;sf2j;s ds
dx, (16)
where b and h denote the width and the thickness of the
rectangular cross-section, respectively.The length of the beam is
equally divided into n segments and the thickness and the width at
everysegment are assumed as cubic spline functions, which can be
expressed, respectively, as follows:
f ix ati x i 1 L
n
3 bti x
i 1 Ln
2 cti x
i 1 Ln
dti ,
gix awi x i 1 L
n
3 bwi x
i 1 Ln
2
cwi x i 1 L dwi i 1; . . . ; n. 17n
-
ARTICLE IN PRESSAll the coefcients of the cubic spline functions
(for the thickness and the width) can bedetermined by the following
conditions:
f ii L
n
hi; gi
i Ln
bi i 1; . . . ; n, (18)
f ii L
n
f i1
i Ln
; gi
i Ln
gi1
i Ln
,
f 0ii L
n
f 0i1
i Ln
; g0i
i Ln
g0i1
i Ln
i 1; . . . ; n 1 ,
f 00ii L
n
f 00i1
i Ln
; g00i
i Ln
g00i1
i Ln
, (19)
f 10 h0; g10 b0,
f 0010 f 00nL 0; g0010 g00nL 0. (20)By using the local spline
functions, the global thickness function hx and the global
widthfunction bx can be expressed as follows:
hx Xni1
f ixuix,
bx Xni1
gixuix, (21)
where
if x 0 then u10 1 and ui0 0 ia1,
if x 2 k 1 Ln
;k L
n
then ukx 1 and uix 0 iak.
Now by using the global thickness and width functions in Eq.
(21), the elements of the mass andthe stiffness matrices shown in
Eqs. (14)(16) can be calculated and the modal analysis employingEq.
(13) can be performed.To verify the accuracy of the modal
formulation (that is derived in this section), three test
problems are solved. Fig. 2 shows the shapes of the three
cantilever beams and the corresponding
H.H. Yoo et al. / Journal of Sound and Vibration 290 (2006)
223241228shape functions. The material and geometric data are given
in Table 1. Numerical results obtained
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ARTICLE IN PRESSh0 x
y
h(x) = hhf
CASE I
H.H. Yoo et al. / Journal of Sound and Vibration 290 (2006)
223241 229by the present method and a commercial code ANSYS are
compared in Table 2. To obtain thepresent results, the beam is
divided into 10 segments in which the thickness is expressed with
aspline function. To obtain the results of ANSYS, the beam is
divided into 50 segments in whichthe thickness is expressed with a
linear function. Table 2 shows that the numerical results
obtainedby the two methods are in good agreement.
h0
h0
L
0
hf
hf
y
x
L
CASE II
x
L
yCASE III
h0xL
h0 hfh(x) +
=
h0x2
L2hf h0h(x) +
=
Fig. 2. Beam thickness variations of three cases of test
problems.
Table 1
Comparison of the rst natural frequency
Angular velocity Case I Case II Case III
Present ANSYS Present ANSYS Present ANSYS
0 2.3409 2.3405 2.6814 2.6812 0.58161 0.58124
10 2.9132 2.9128 3.2408 3.2407 1.7707 1.7698
20 4.1649 4.1651 4.4847 4.4852 3.3515 3.3457
30 5.6349 5.6371 5.9540 5.9562 4.9526 4.9374
(unit: rad/s).
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Youngs modulus E GPa 70Density r kg=m3 1:2 1033. Formulation of
optimization problems and numerical results
Length L m 10Width b m 0.3Fixed end height h0 m 0.3 0.3 0.1Free
end height hf m 0.3 0.1 0.3
Table 3
Material and geometric properties of the beam for optimization
problems
Description Data
Youngs modulus E GPa 69.0Density r kg=m3 2:71 103Length L m
0.4Table 2
Material and geometric properties of the beam for test
problems
Description Case I Case II Case III
H.H. Yoo et al. / Journal of Sound and Vibration 290 (2006)
223241230The natural frequencies of rotating beams can be
determined from the angular speed as well asthe thickness and the
width that consist of the cross-section. Thus, the natural
frequencies can beexpressed as follows:
ok okO; X , (22)where ok denotes the kth natural frequency, O
denotes the angular speed of the rotating beam,and X denotes the
stage values of the thickness and the width, which are employed as
designvariables. Thus, if a beam is divided into n segments, X can
be expressed as follows:
X h0; h1; . . . ; hn; b0; b1; . . . ; bnT, (23)
where hi and bi represent the stage values of the thickness and
the width from the xed end to thefree end of the cantilever beam,
respectively. The material and geometric properties of the beamare
given in Table 3. To obtain the natural frequencies, ve bending
modes are employed.
3.1. Problem to find the range of the first natural frequency of
a rotating beam
The rst problem is to nd the range of the rst natural frequency
of a rotating beam when thefollowing constraints are given. The
total volume of the beam needs to be not larger thanthe initial
volume and the thickness and the width have minimum values. To
solve the problem,
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the objective function and the constraints are formulated as
follows:
Min or Max o1Os; X
s:t:
Z L0
hX ;xbX ; xdxpLhinibini,
hX ;xXhmin 0pxpL,bX ;xXbmin 0pxpL, 24
where Os denotes the angular speed; hini 0:002m and bini 0:035m
denote the initial thicknessand the initial width; and hmin 0:001m
and bmin 0:0175m denote the minimum thicknessand the minimum width
of the cross-section. For this problem, the hub radius is given 0
and thelength of the beam is divided into 10 segments. Thus, the
design variables consist of 22 elements.Fig. 3 shows the minimum
and the maximum rst natural frequency loci of the rotating beam
when the angular speed increases from 0 to 300 rad/s. Therefore,
the two loci embrace the possibleregion of the rst natural
frequency of the rotating beam. To obtain the results in Fig. 3,
two
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H.H. Yoo et al. / Journal of Sound and Vibration 290 (2006)
223241 231optimization problems (to nd the minimum and the maximum
natural frequencies) for theangular speed of 0 rad/s, are rst
solved with the given initial design variables. The optimumvalues
of design variables are then employed as the initial values to
solve the optimizationproblems for the angular speed of 0.1 rad/s.
The same procedure continues until the angular speedreaches 300
rad/s. Thus, 3001 sets of optimization problems need to be solved
to obtain theminimum and the maximum loci, respectively. Table 4
shows a typical set of numerical results forthe optimization
problem. The initial and the optimal values of the design variables
and theobjective function (to nd the rst minimum natural frequency)
for the angular speed of 0 rad/sare shown in the table. Fig. 4 also
shows the convergence history of the objective function.Figs. 5 and
6 represent the thickness and the width of the beam versus the
length of the beam
for the minimum and the maximum rst natural frequency results,
respectively. The thickness andthe width variations along the beam
length for the angular speed of 0 and 300 rad/s are shown in
0 50 100 150 200 250 3000
100
200
300
400
500
600
700 Locus of the maximum 1st natural frequency Locus of the
minimum 1st natural frequency
Nat
ural
freq
uenc
y (ra
d/s)
Angular speed (rad/s) Fig. 3. Two loci that embrace the rst
natural frequency range.
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ARTICLE IN PRESS
Table 4
Initial and optimal values of the design variables and the
objective function
Initial values Optimal values
Design variables (m)
h0 0.002 0.00100
h1 0.002 0.00100
h2 0.002 0.00102
h3 0.002 0.00114
h4 0.002 0.00144
h5 0.002 0.00195
h6 0.002 0.00275
h7 0.002 0.00389
h8 0.002 0.00542
h9 0.002 0.00740
h10 0.002 0.00989
b0 0.035 0.01750
b1 0.035 0.01811
b2 0.035 0.01873
b3 0.035 0.01938
b4 0.035 0.02006
b5 0.035 0.02076
b6 0.035 0.02148
b7 0.035 0.02223
b8 0.035 0.02301
b9 0.035 0.02382
b10 0.035 0.02465
Objective function (rad/s)
64.019 13.190
0 20 40 60 80 1000
10
20
30
40
50
60
70
Nat
ural
freq
uenc
y (ra
d/s)
Iteration number
Objective function historywhen angular speed is 0 rad/s
Fig. 4. Convergence history of the objective function.
H.H. Yoo et al. / Journal of Sound and Vibration 290 (2006)
223241232
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H.H. Yoo et al. / Journal of Sound and Vibration 290 (2006)
223241 2331
2
3
4
5
6
7
Hal
f the
thic
knes
s (m
m)
Minimum result (s = 0 rad/s) Minimum result (s = 300 rad/s)
Thickness limitthese gures. The minimum frequency results show that
both the thickness and the widthgradually increase as the
cross-section moves to the free end. The maximum frequency
results,however, show that the width xes to the minimum value, bmin
while the thickness varies with aninteresting curve. The thickness
curve has three peaks that decrease as the cross-section moves
tothe free end. These gures also show that the cross-section shapes
that minimize and maximize therst natural frequencies do not vary
even if the angular speed varies. Figs. 7 and 8 show theoverall
pictures of the beams for the minimum and the maximum frequency
results, respectively.To show cross-sections of the beams in
detail, the thickness and the width are amplied two timescompared
to the length of the beam.It is well known that (as the angular
speed increases) the natural frequencies of cantilever beams
with larger hub radius increase faster than those with smaller
hub radius. To investigate the effectof the hub radius on the
optimal shape of rotating beam, problems with different hub
radius
0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36 0.400
Length (m)(a)
0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36 0.400
5
10
15
20
25
Hal
f the
wid
th (m
m)
Length (m)
Minimum result (s = 0 rad/s) Minimum result (s = 300 rad/s)
Width limit
(b)
Fig. 5. Variation of the thickness and the width along the beam
length that minimize the rst natural frequency. (a)
Half the thickness, (b) half the width.
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ss (2
3
4
Hal
f the
thic
kne5
6
7
mm)
Maximum result (s = 0 rad/s) Maximum result (s = 300 rad/s)
Thickness limit
H.H. Yoo et al. / Journal of Sound and Vibration 290 (2006)
223241234values are solved. Fig. 9 shows the thickness and the
width of the beam (that maximize the rstnatural frequency) with
three different radius values. The results show that the optimal
shapes forthe three different radius values are identical. So, for
other design problems, hereinafter, the hubradius of the rotating
beam is xed as 0.
3.2. Problem with the fixed stationary first natural frequency
constraint
In many structural design problems, the stationary rst natural
frequency (when the angularspeed is zero) often needs to be
constrained to the initially given value. So one more constraint
(asa design requirement) is added to the previous problem. Thus,
the objective function and
0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36 0.400
1
Length (m)
0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36 0.400
5
10
15
20
25 Maximum result (s = 0 rad/s) Maximum result (s = 300 rad/s)
Width limit
Hal
f the
wid
th (m
m)
Length (m)
(a)
(b)
Fig. 6. Variation of the thickness and the width along the beam
length that maximize the rst natural frequency.
(a) Half the thickness, (b) half the width.
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H.H. Yoo et al. / Journal of Sound and Vibration 290 (2006)
223241 235constraints are now given as follows:
Min or Max o1Os; X s:t: o10; X o10; X 0 0,Z L
0
hX ;xbX ; xdxpLhinibini,
hX ;xXhmin 0pxpL,bX ;xXbmin 0pxpL. 25
Fig. 7. Beam shape that minimizes the rst natural frequency.
Fig. 8. Beam shape that maximizes the rst natural frequency.
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ARTICLE IN PRESS2
3
4
5
6
7 r = 0.0 r = 0.5L r = 2.0L Thickness limit
Hal
f the
thic
knes
s (m
m)
H.H. Yoo et al. / Journal of Sound and Vibration 290 (2006)
223241236Fig. 10 shows the minimum and the maximum rst natural
frequency loci of the rotating beamwhen the angular speed increases
from 0 to 100 rad/s. Thus, the two loci embrace the possibleregion
of the rst natural frequency. As shown in the gure, the rst natural
frequency isconstrained to a value when the angular speed is 0
rad/s.Figs. 11 and 12 show the thickness and the width of the beam
versus the length of the beam for
the minimum and the maximum rst natural frequency results. The
thickness and the widthvariations along the beam length with the
angular speed of 100 rad/s are shown in these gures.Figs. 13 and 14
show the overall pictures of the beams for the minimum and the
maximumfrequency results, respectively. The thickness and the width
are again amplied two timescompared to the beam length.
0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36 0.400
1
Length (m)
0
5
10
15
20
25
Hal
f the
wid
th (m
m)
r = 0.0 r = 0.5L r = 2.0L Width limit
0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36 0.40Length
(m)(b)
(a)
Fig. 9. Variation of the thickness and the width along the beam
length that maximize the rst natural frequency for
three cases of hub radius. (a) Half the thickness, (b) half the
width.
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ARTICLE IN PRESS130140150160170
Locus of the maximum 1st natural frequencyLocus of the minimum
1st natural frequency
y (ra
d/s)
H.H. Yoo et al. / Journal of Sound and Vibration 290 (2006)
223241 2373.3. Problem to find a beam shape having a specified
first natural frequency at a specific angularspeed with the fixed
stationary first natural frequency constraint
As a typical design problem, the rst natural frequency of a
rotating beam at a specic angularspeed (usually this is the
operating speed) is rst determined to avoid undesirable problems
likeresonance phenomena. Then the beam shape that satises the
condition needs to be found. To
0 20 40 60 80 10060708090
100110120
Angular speed (rad/s)
Nat
ural
freq
uenc
Fig. 10. The rst natural frequency range for the optimization
problem including the rst natural frequency constraint
at zero angular speed.
0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36 0.400
1
2
3
4
5 Maximum result Minimum result Thickness limit
Hal
f the
thic
knes
s (m
m)
Length (m)
Fig. 11. Thickness variations that minimize and maximize the rst
natural frequency (for the optimization problems
with an added constraint).
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H.H. Yoo et al. / Journal of Sound and Vibration 290 (2006)
22324123810
15
20
25
alf t
he w
idth
(mm)
Maximum result Minimum result Width limitsolve such a design
problem, the following formulation can be employed:
Min o1OG;X oG2s:t: o10; X o10;X 0 0,Z L
0
hX ; xbX ; xdxpLhinibini,
hX ; xXhmin 0pxpL,bX ; xXbmin 0pxpL, 26
0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36 0.400
5
H
Length (m)
Fig. 12. Width variations that minimize and maximize the rst
natural frequency (for the optimization problems with
an added constraint).
Fig. 13. Beam shape that minimizes the rst natural frequency
while satisfying the added constraint of the rst natural
frequency at zero angular speed.
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H.H. Yoo et al. / Journal of Sound and Vibration 290 (2006)
223241 239where OG and oG are the specic angular speed and the
corresponding rst natural frequency. Theobjective function is
different from the previous problem but the constraint equations
are same asthe previous one.Fig. 15 shows two loci and a point: the
maximum rst natural frequency locus, the minimum
rst natural frequency locus, and the point that satises o1OG;X
oG 0. For this problem,OG is given as 50 rad/s and oG is given as
90 rad/s. Of course, only the natural frequency that islocated
between the minimum and the maximum frequency loci may be required
for the designproblem. Fig. 16 shows the variations of the
thickness and the width of the beam that has thespecic natural
frequency at the specic angular speed and Fig. 17 shows the overall
picture of the
Fig. 14. Beam shape that maximizes the rst natural frequency
while satisfying the added constraint of the rst natural
frequency at zero angular speed.
0 20 40 60 80 10060708090
100110120130140150160170
Locus of the maximum 1st natural frequency Locus of the minimum
1st natural frequencySpecified natural frequency condition
Nat
ural
freq
uenc
y (ra
d/s)
Angular speed (rad/s)
Fig. 15. A specied natural frequency condition at a non-zero
angular speed that is located between the maximum and
the minimum natural frequency loci.
-
ARTICLE IN PRESS
hick
n
e w
id1
2
Hal
f the
t
5
10
Hal
f th3
4
5
ess
(mm)
Thickness shape Width shape
15
20
25
th (m
m)
H.H. Yoo et al. / Journal of Sound and Vibration 290 (2006)
223241240beam. The thickness and the width of the beam
cross-section is again amplied two timescompared to the length of
the beam.
4. Conclusions
In this study, an optimization method is employed to nd the
cross-section shape variations ofrotating cantilever beams that
satisfy some specic modal characteristics. The beam is divided
intomultiple segments and the thickness and the width are assumed
as cubic spline functions at thesegments. The stage values of the
thickness and the width are employed as design variables and
Fig. 17. Beam shape that satises a specied modal characteristic
at a non-zero angular speed.
0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36 0.400
Length (m)0
Fig. 16. Variations of the thickness and the width that satises
a specied modal characteristic at a non-zero angular
speed.
-
optimization problems to nd the design variables are formulated.
Numerical results show thatthere exist specic cross-section shape
variations that satisfy certain modal characteristicrequirements.
It is also found that the angular speed and the hub radius little
inuence the optimalshapes of the rotating beams.
[12] H. Yoo, S. Shin, Vibration analysis of rotating cantilever
beams, Journal of Sound and Vibration 212 (1998)
ARTICLE IN PRESS
H.H. Yoo et al. / Journal of Sound and Vibration 290 (2006)
223241 241807828.
[13] D. Goldberg, Genetic Algorithms in Search, Optimization and
Machine Learning, Addison-Wesley, Longman,
Reading, MA, 1989.
[14] G. Vanderplaats, DOTDesign Optimization Tools User Manual,
Vanderplaats Research & Development, 1999.Acknowledgements
This research was supported by the Innovative Design
Optimization Technology EngineeringResearch Center through the
research fund, for which the authors are grateful.
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Modal analysis and shape optimization of rotating cantilever
beamsIntroductionDerivation of the modal equations of a rotating
cantilever beamFormulation of optimization problems and numerical
resultsProblem to find the range of the first natural frequency of
a rotating beamProblem with the fixed stationary first natural
frequency constraintProblem to find a beam shape having a specified
first natural frequency at a specific angular speed with the fixed
stationary first natural frequency constraint
ConclusionsAcknowledgementsReferences